Supplementary Information for Reconstruction of Transcriptional Dynamics from Gene Reporter Data Using Diﬀerential Equations

Supplementary information for Reconstruction of transcriptional dynamics from gene reporter data using diﬀerential equations.

B. Finkenst¨adt,E. A. Heron, M. Komorowski, K. Edwards, S. Tang, C. V. Harper, J. R. E. Davis, M. R. H. White, A. J. Millar and D. A. Rand

Abstract This is supplementary information for the main paper Reconstruction of transcriptional dynamics from gene reporter data using diﬀerential equations. The main paper is referred to as MP.

Scaled Model

Suppose that we measure M,P indirectly through variables M˜ (t) = sM M(t) for the mRNA and P˜(t) = sP P (t) for the reporter protein. Re-formulating model (1) in MP gives a scaled model

dM/dt˜ =τ ˜(t; θτ ) − δM M˜ (t)

dP˜ /dt =α ˜M˜ (t) − δP P˜(t), (1)

sP where the transcription functionτ ˜(t; θτ ) = sM τ(t; θτ ) and the translation rateα ˜ = α are now functions sM of the unknown scaling coeﬃcients sM , sP . Obviously, the functional forms of (1) and model (1) in MP are identical. If we set M = M,P˜ = P˜ , τ =τ ˜ and α =α ˜ then (1) in MP denotes the scaled model.

Diﬀusion Approximation

Let pt(M,P ) denote the probability that at time t system is in the state (M,P ). The evolution of the joint probability is described by the chemical master equation of the form (see (Thattai and van Oudenaarden 2001) for derivation) dp (M, P, t) t = τ(t)(p (M − 1,P ) − p (M,P )) + αM(p (M,P − 1) − p (M,P )) (2) dt t t t t +δM (pt(M + 1,P )(M + 1) − pt(M,P )M) + δP (pt(M,P + 1)(P + 1) − pt(M,P )P ). In order to obtain a diffusion approximation of (2) we replace increments on the right hand of the above with their second order Taylor expansions. This gives the Fokker-Planck equation dp (M,P ) ∂ t = − (τ(t) − δ M)p (M,P ) dt ∂M M t ∂ − (αM − δ P )p (M,P ) ∂P P t 1 ∂2 + (τ(t) + δ M)p (M,P ) 2 ∂M 2 M t 1 ∂2 + (αM + δ P )p (M,P ). 2 ∂P 2 P t This method is called Ω (size) expansion and gives valid approximation for system of large volume (see (Van Kampen 2006; Golightly and Wilkinson 2005) for details). The Fokker-Planck equation describes the evolution of the probability densities of the stochastic process governed by the Itôdiffusion (Gardiner 1985) p dM = (τ(t) − δM M)dt + τ(t) + δM MdWr (3) p dP = (αM − δP P )dt + αM + δP P dWp,

1 2 Supplementary information

where dWr, dWp are increments of independent Wiener processes and thus their joint distribution is bivariate normal as stated in (*) of MP. Higham (2001) gives an accessible algorithmic introduction to stochastic diﬀerential equations as in (3) and Wiener processes.

Supplementary information for Case study 1

Experiment LUC reporter constructs fused to the CAB2 promoter CAB2:LUC, have allowed the characterization of CAB2 expression during high resolution imaging time-courses (Millar et al. (1995)). We used the well characterized induction of CAB2 expression by light to study the activity of LUC in plants. Arabidopsis seed containing the CAB2:LUC reporter gene were given a 6h white light pulse to induce germination and then grown in constant darkness for 4 days. Seedlings were grown in darkness to reduce the basal level of CAB2:LUC expression. Since induction of CAB2 by light is gated by the clock, occurring maximally during the early part of the day (Millar and Kay (1996)), the seedlings were entrained under temperature cycles of 12h at 24 degrees C followed by 12h at 18 degrees C, allowing us to target the light pulse to the relevant time of the day. At dawn on the 5th day the temperature cycles were stopped and the plants were maintained in darkness at 22 degrees C. They were also transferred to liquid media, containing 1mM Luciferin to ensure that the substrate did not become limiting to LUC activity. At subjective dawn on the 6th day (24h after the transfer to constant temperature; referred to as time 0 in the text and figures), the seedlings were given a 20min red light pulse to induce CAB2 expression. Samples were harvested at the indicated time-points and total-RNA and -protein was extracted. Steady state levels of LUC mRNA were measured by Quantitative PCR (Q-PCR) and an in vitro LUC assay (Promega, Madison,WI, USA) was used to measure LUC activity in the protein samples. Concurrently, red light pulsed seedlings were also imaged for LUC activity using light sensitive cameras (Millar et al. (1995)). This allows the measurement of LUC activity within the same seedlings throughout the entire experiment, whereas the in vitro LUC assays and Q-PCR experiments necessarily sacrificed different samples for each time-point. Model Assuming that molecular populations scale differently with the Q-PCR, in vitro and in vivo imaging data we use the following equations based on model (1) in MP

dM = τ(t) − δ M(t), (4) dt M dP = α M(t) − δ P (t) + c , (5) dt P P P dP v = α M(t) − δ P (t), (6) dt Pv P v

The additional variable Pv represents the protein dynamics measured via the in vitro LUC protein assays. Both protein equations have identical degradation rates δP and translation proportional to M(t) but with diﬀerently scaled translation rates αP and αPv . Preliminary estimations also showed that the observed near zero levels of the control imaging data are only compatible with the higher control levels of the in vitro protein and RT-PCR mRNA if a constant cP is added to the imaging data.

Supplementary information for Case study 2

Experiment Circadian regulation is normally tested by entraining the organism to 12h light: 12h dark cycles, then transferring the organism to constant conditions. Transgenic Arabidopsis seed were sterilised and grown as described previously (Gould et al. 2006), for 4 days at 22o C in Sanyo MLR350 environmental test chambers (Sanyo, Osaka, Japan) under photoperiods of 75µ moles.m-2s-1 cool white ﬂuorescent light. Seedlings were then transferred to Percival I-30BLL growth chambers (CLF Plant Climatics, Emersacker, Germany) at dawn on the 5th day and grown at 22o C under an equal mix of Red and Blue LEDs at 20-30µ moles.m-2s-1, with 18h light: 6h dark photoperiods. In the data shown in Figure 2 of MP, time 0 is the time of lights-on on the 7th day of growth. CCA1:LUC+ plants have been described in (Doyle et al. 2002). Luciferase imaging was carried out as previously described (Gould et al. 2006) using Hamamatsu C4742-98 digital cameras Finkenst¨adtet al. 3

Figure 1: Plot of data for case study 1. Left: LUC mRNA Q-PCR, middle: imaging of LUC protein, right: in-vitro LUC assay, top row: experiment with red light stimulus during ﬁrst 20 minutes, bottom row: un-stimulated control experiments. There are three replicates. A big dot corresponds to an observed data point.

operating at −75o C under control of Wasabi software (Hamamatsu Photonics, Hamamatsu City, Japan). Bioluminescence levels were quantified using Metamorph software (MDS, Toronto, Canada). Experiments included 22 individuals of each genotype and were replicated 4 or more times. For Q-PCR experiments, wild type Wassilewskija (Ws) seedlings were grown for 7 days in Percival Growth chambers under experimental photoperiods of 60-65µMolesm-2s-1 cool white fluorescent light. Seedlings were harvested, RNA was extracted and reverse transcribed as described previously (Locke et al. 2005). Quantitative PCR was carried out in 384-well format using SYBR Green JumpStart Taq ReadyMix (Sigma, Gillingham, UK) in techni- cal triplicate with a LightCycler 480 instrument (Roche, UK), using the Relative Quantification function to measure mRNA abundance. Expression values were normalised against ACTIN 2 (ACT2). ACT2 and CCA1. PCR primers have previously been described (Locke et al. 2005). Model The model for case study 2 is

dM g = τ(t) − δ M (t), (7) dt Mg g dM = τ(t) − δ M(t), (8) dt M dP = αM(t) − δ P (t), (9) dt P

where (8) and (9) are as in model (1) of MP, describing the dynamics of the luciferase reporter mRNA and protein, respectively. Equation (7) formulates the transcription and degradation of the native gene CCA1 mRNA for which we have coarse Q-PCR data. Equations (7)-(9) are at the population level. We assume

that the observed variables are proportional to Mg and P with scaling factors sMg and sP , respectively, whilst M is unobserved. Equations 7 and 9 represent an equivalent parameterization of the scaled variables

withτ ˜(t) = sMg τ(t) andα ˜ = (sP /sMg )α replacing the transcription and translation coeﬃcient, respectively. For ease of notation we re-use τ(t) and α. 4 Supplementary information

Parameter average r1 r2 r3 αM 10.43 (0.232) 9.62 (0.179) 8.36 (0.538) 25.38 (1.43) δM 1.542 (0.019) 1.726 (0.044) 1.417 (0.121) 3.526 (0.315) (half-life) 0.45 h 0.4 h 0.49 h 0.2 h µΓ 2.008 (0.011 ) 2.101 (0.014) 1.902 (0.045) 2.362 (0.0289) σΓ 0.631 (0.013) 0.692 (0.014) 0.686 (0.039) 0.723 (0.0217) τ 0.012 (0.001) 0.014 (0.001) 0.014 (0.002) 0.013 (0.002) αP 25.07 (0.386) 34.90 (0.555) 23.02 (1.417) 24.83 (1.78) δP 0.305 (0.0045) 0.286 (0.0040) 0.272 (0.010) 0.365 (0.0093) (half-life) 2.27 h 2.42 h 2.5 h 1.9 h αP v 2.178 (0.107) 2.141 (0.210) 2.534 (0.222) 2.152 (0.183) cP -1.868 (0.198) -2.637 (0.144) -1.763 (0.208) -2.121 (0.388) σM 0.159 (0.0026) 0.254 (0.0018) 0.134 (0.0069) 0.171 (0.005) σP 0.183 (0.0133) 0.182 (0.0125) 0.286 (0.0287) 0.331 (0.030) σP v 0.364 (0.0211) 0.769 (0.0222) 0.562 (0.0129) 0.442 (0.034)

Table 1: Posterior means and standard deviations of all estimated parameters where the red light pulse model was ﬁtted to average data and to single replicate data sets denoted by r1, r2, r3. Estimated rates are per hour. Degradation rates are translated into half-life as follows: half-life (in hours)=ln(2) /degradation rate (per hour). σM , σP , σP v give estimated posterior standard error for each model equation.

MCMC algorithm for inference using SDE approach

(i) ∗ ∗ 1. Set iteration counter i = 0. Initialise parameters θ and all bridges Mg and P . 2. Set i = i + 1

3. Update τ (i), τ (i) and δ(i) in one block. Use individual random walk Metropolis proposals and either on oﬀ Mg all are accepted or all are rejected. If the proposals are accepted update M (i) and τ (i).

(i) 4. Similarly, update Sw in one block using a random walk Metropolis step.

5. Update s(i) using a random walk Metropolis step. Mg

(i) (i) (i) 6. Update δM using a random walk Metropolis step. If the proposal is accepted update M and τ .

(i) (i) 7. Update M0 using a random walk Metropolis step. Update M .

(i) (i) 8. Update α and δP in one block using an independence sampler.

(i) 9. Update sP using a random walk Metropolis step. ∗ 10. Sample Mg bridges (using the method in Elerian et al. (2001)). 11. Similarly, update P ∗ bridges.

12. Repeat from Step 2 until a suﬃcient sample from the converged chains has been obtained.

Supplementary information for Case study 3

Experiment GH3 rat pituitary cells stably transfected with 5kb human prolactin promoter destabilized EGFP reporter construct (hPRL-d2EGFP) were seeded onto 35 mm glass coverslip-based dishes (IWAKI, Japan) and cul- tured in 10 % FCS for 24 h prior to imaging. Cells were transferred to the stage of a Zeiss Axiovert 200 equipped with an XL incubator (maintained at 37C, 5 % CO2, in humid conditions) and images were obtained Finkenst¨adtet al. 5

Parameter SDE ODE

δMg 0.426 (0.0043) 0.313 (0.0273) δM 1.54 (0.019) 1.42 (0.101) α 0.095 (0.0090) 0.113 (0.0050) δP 0.072 (0.0057) 0.075 (0.0018) M(0) 19508 (13880) 13487 (6950) τon 21401 (331) 17865 (1366) τoﬀ 651 (33.85) 0 pd 2.35 (0.05) 4.39 (0.58)

sMg (SDE), σMg (ODE)) 24.99 (0.013) 7560 (188) sP (SDE), σP (ODE) 188.6 (8.922) 290 (11.33) s1 0.56 (0.030) 0.26 (0.09) s2 21.11 (0.040) 21.85 (0.11) s3 25.07 (0.046) 23.76 (0.16) s4 42.12 (0.055) 42.74 (0.19) s5 50.92 (0.052) 50.28 (0.19) s6 66.92 (0.063) 66.53 (0.18) s7 71.80 (0.133) 73.66 (0.26) s8 86.96 (2.10) 89.55 (0.22)

Table 2: Posterior mean and standard error estimates of parameters and switch-times for the experimental data using the SDE and mean ODE approach in case study 2. using a Fluar x20, 0.75 numerical aperture (Zeiss), air objective. Excitation of d2EGFP was performed using an Argon ion laser at 488nm. Emitted light was captured through a 505-550 nm bandpass ﬁlter from a 545 nm dichroic mirror. Images were captured every 15 min. 5 M forskolin and 0.5 M BayK 8644 were added directly to the dish at the start of the experiment. Data was captured and analysed using LSM510 software with consecutive autofocus. Analysis was performed using Kinetic Imaging software AQM6. Regions of interest were drawn around each single cell and mean intensity data was measured 108 times in 15 minutes intervals giving a total of 27 hours of data (see Figure (3) in MP). Transformation of parameters To reduce correlation and improve convergence of the chain we re-parameterized the model in case study 3 as follows

h(Θ,M,P ) = (Θˆ , M,P˜ ) (10) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = (θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8, θ9, M,P˜ ) (11)

= (log(δM ), log(δP ), log(sP αb0), log(sP α), log(α),

log(α sP b4), log(b3), log(b1), log(b2), M,P˜ ), where M˜ = sM αM − δP P .

Algorithm for inference

1. Set iteration counter i = 0. Initialise all parameters Θˆ (i), hidden process M˜ and bridges P ∗. 2. Set i = i + 1 ˆ(i) ˆ(i) ˆ(i) 3. Update (θ1 , θ3 , θ6 ) in one block using multivariate normal proposals and either all are accepted or all are rejected. 4. Separately update each remaining component of Θ(i) using random walk proposals. 5. Update P ∗ bridges (using the method in Elerian et al. (2001)) 6. Similarly, update the latent process M˜ . 6 Supplementary information

7. Repeat from Step 2 until a suﬃcient sample from the converged chains has been obtained.

value prior Simulation Experiment δM 0.44 Γ(0.44,0.02) 0.56 ( 0.36 - 0.92 ) 0.45(0.26 - 0.82 ) δP 0.52 Γ(0.52,0.02) 0.59 (0.38 - 0.89) 0.71 ( 0.45 - 1.09 ) α 20 Exp(100) 16.97 ( 6.54 - 78.98 ) 0.46 ( 0.14 - 1.51 ) sP 0.2 Exp(1) 0.17 ( 0.09 - 0.3 ) 2.11 ( 1.24 - 3.56 ) b0 23 Exp(100) 40.48 ( 9.11 - 136.3 ) 112.7 ( 29.52 - 364.8 ) b1 10 Exp(50) 31.52 ( 7.08 - 83.64 ) 54.84 ( 21.67 - 97.14 ) b2 30 Exp(50) 22.83 ( 5.82 - 60.13 ) 59.43 ( 19.08 - 130.6 ) b3 5 Exp(7) 7.22 ( 4.35 - 9.55 ) 13.78 ( 11.08 - 16.05 ) b4 5 Exp(100) 6.4 ( 1.55 - 23.68 ) 7.39 ( 0.25 - 30.75 ) M0 15 Exp(70) 23.11 ( 4.8 - 66.92 ) 31.79 ( 7.81 - 97.87 )

Table 3: Posterior inference results for case study 3. Parameter values used in simulation study. Priors, posterior medians and 95% credibility intervals inferred from both simulated and experimental data. Rates are per hour. Γ(µ, σ2) denotes gamma distribution with mean µ and variance σ2.

Updating Bridges

Suppose data Y = (Y (t1), ..., Y (tT )) are provided at sampling intervals that are too coarse to allow parameter estimation in the SDE approach without bridging. For example, LUC protein imaging data may be available every 30 minutes while for artificial stochastic process data from simulated clock models we find that a small enough time interval for the normal approximation in (*) of MP to produce reasonably accurate parameter estimates is 0.1 hour. The methods applied in this study make use of established strategies developed for nonlinear stochastic differential equations (Pedersen 1995; Kim et al. 1998; Eraker 2001; Elerian et al. 2001; Durham and Gallant 2002). The basic idea is to augment the observed data by introducing a number of latent data points (called bridges) Y ∗ in-between the measurements. The bridges are constructed so that the data together with the bridges (augmented data) give a time series with interval length ∆ti = 0.1 h which we know from simulation studies allows for accurate parameter estimation. To provide an estimate of the parameters θ from sparsely sampled data, we use MCMC to sample from the joint posterior f(θ, Y ∗|Y ) of the parameters θ and the auxiliary variables Y ∗ given the data Y , using the fact that, by Bayes’ theorem,

∗ ∗ f(θ, Y |Y ) ∝ LSDE(Y ,Y |θ)π(θ) (12)

∗ where, as before, π(θ) denotes the prior distribution on θ and LSDE(Y ,Y |θ) is the approximated augmented likelihood. This is achieved by sampling in turn from the full conditional densities of θ|Y ∗,Y and Y ∗|θ, Y (Tanner and Wong (1987)). The general structure of the algorithm that we employ is thus as follows:

1. Initialise Y ∗ by constructing linear bridges between each of the given data points. The parameters θ are initialised as usual.

∗ ∗ 2. Sample Yi from Yi |Y (ti),Y (ti+1), θ for i = 1, 2,...,T − 1. The two samples constitute a full set of imputed data Y ∗.

3. Sample θ from θ|Y,Y ∗, i.e. use the fully augmented data to update the parameter vector.

4. Repeat steps 2 and 3 until the required sample is obtained after the chain has converged.

Updating the parameter vector in step 3 is quite straightforward as for a given fully augmented time path the constant rate approximation for (*) in MP is valid and the inference problem is the same as for a ﬁnely ∗ sampled time path. To sample Yi in step 2 we use the bridging methodology suggested by Elerian et al. (2001) which has proved very satisfactory but it should be noted that there exist various other available methods for bridging (see Durham and Gallant (2002) for a survey) that may also be used for this kind of Finkenst¨adtet al. 7

problem. We now brieﬂy describe the bridge building part (step 2) of the algorithm using the Elerian et al. (2001) sampler. Consider a general SDE of the form:

dy(ti) = µ(y(ti), ti, θ)dt + σ(y(ti), ti, θ)dW, (13)

∗ ∗ where y could be, for example, mRNA (M) or protein (P ). We denote y(ti) by yi and y (τi,j) by yi,j. Consider any two consecutive observations (yi, yi+1), the observed time series being given by y = (y1, y2, . . . , yT ). We want to impute a bridge of F auxiliary data points between the pair yi and yi+1 at times (τi,1, . . . , τi,F ), ∗ ∗ ∗ where τi,j+1 − τi,j = ∆ (= 0.1 h for example) for all j = 1,...,F − 1. Let yi = (yi,1, . . . , yi,F ) denote the ∗ ∗ ∗ auxiliary bridge and let y = (y1 , . . . , yT −1) denote all the auxiliary bridges. ∗ We know that yi is conditionally independent of the other bridges, given (yi, yi+1, θ). Thus

T −1 ∗ Y ∗ f(y |y1, y, θ) = f(yi |yi, yi+1, θ), i=1

and

F ∗ Y ∗ ∗ f(yi |yi, yi+1, θ) ∝ f(yi,j+1|yi,j, θ), j=0 F Y ∗ ∗ ∗ 2 ∗ ∝ Φ(yi,j+1 − yi,j; µ(yi,j, τi,j, θ)∆, σ (Yi,j, τi,j, θ)∆), j=0

where Φ is the Normal density. Given two data points we construct a bridge of length F between them, but sampling a bridge of length F is ∗ not recommended because it is diﬃcult to sample a high-dimensional yi in one block. Instead we construct ∗ ∗ sub-bridges of length m. A sub-bridge of m auxiliary data points starts at yi,k and ends at yi,k+m−1.

∗ ∗ ∗ ∗ yi(k,m) = (yi,k, yi,k+1, . . . , yi,k+m−1), k = 1, m − 1, 2m − 1,... (14)

∗ ∗ The density conditioned on the two points at either end of this sub-bridge, yi,k−1, yi,k+m, is given by

k+m ∗ ∗ ∗ Y ∗ ∗ ∗ 2 ∗ f(yi(k,m)|yi,k−1, yi,k+m, θ) ∝ Φ(yi,j+1 − yi,j; µ(yi,j, τi,j, θ)∆, σ (yi,j, τi,j, θ)∆). (15) j=k−1

We sample each of the sub-bridges of length m in sequence and accept or reject each of them using the ∗ ∗ ∗ Metropolis-Hastings algorithm (Chib and Greenberg (1995)). Let q(yi(k,m)|yi,k−1, yi,k+m, θ) denote the pro- ∗ posal density. Suppose that at each iteration n of our MCMC algorithm, the sub-bridge yi(k,m) is given by ∗(n) ∗ ∗ ∗ yi(k,m). We propose a new sub-bridge w ∼ q(yi(k,m)|yi,k−1, yi,k+m, θ). The new sub-bridge is then accepted with probability:

The proposal density q(.|.) is chosen to be a multivariate Normal approximation of the target density at the mode. The location of q(.|.) is given by the mode of the target density obtained by a few Newton-Raphson steps and the dispersion is given by the negative of the inverse Hessian evaluated at the mode. This is a multi-dimensional independence sampler, as the proposal distribution q(.|.) does not depend on the current value of the chain. Elerian et al. (2001) give analytic functions for both the gradient and negative Hessian for the type of stochastic diﬀerential equation we are considering, removing the need to approximate these functions. Step 3, i.e. updating the parameters, is carried out as usual with the augmented data (Y,Y ∗) being treated in the same way as if we had ﬁne data. 8 Supplementary information

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